A school club uses a telephone chain to inform members of changes in plans. The club president calls 2 members, each of whom calls 2 members, and so on. The process must be repeated 5 times. How many members are in the club?

Since each person calls 2 persons each, there is an exponential increase of 2 for each level. Since it starts with the president as 1 person, and increases exponentially by 2 for each level of phone call, we have a geometric progression of first term, a = 1 and common ratio r = 2. Since the process of phone calls must be repeated 5 times, our last term is 2⁵. So, our geometric progression is 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ = 1 + 2¹ + 2² + 2³ + 2⁴ + 2⁵

We now sum the geometric progression

The sum of a geometric progression with r > 1 is

S = a(rⁿ - 1)/(r - 1)

substituting r = 2, a = 1 and n = 6 since we have 6 terms, we have

S = 1(2⁶ - 1)/(2 - 1)

= (64 - 1)/1

= 63

So, we have 63 members in the club.

We could have also found this by summing up individually the terms in the series. We have 1 + 2¹ + 2² + 2³ + 2⁴ + 2⁵ = 1 + 2 + 4 + 8 + 16 + 32 = 63